Find Two Points, Which Divides The Line Segment Joining A (3, 5) And B (-3, 2) Into Three Equal Parts.
Introduction
In this problem, we are given two points A (3, 5) and B (-3, 2) and we need to find two points that divide the line segment joining these two points into three equal parts. This is a classic problem in geometry and can be solved using the concept of section formula.
Understanding the Problem
To solve this problem, we need to understand the concept of section formula. The section formula states that if a point P (x, y) divides the line segment joining the points A (x1, y1) and B (x2, y2) in the ratio m:n, then the coordinates of P are given by:
(x, y) = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))
Applying the Section Formula
In this problem, we need to find two points that divide the line segment joining A (3, 5) and B (-3, 2) into three equal parts. This means that the ratio in which the points divide the line segment is 1:2.
Using the section formula, we can find the coordinates of the first point that divides the line segment in the ratio 1:2. Let the coordinates of this point be (x1, y1).
(x1, y1) = ((1*(-3) + 23)/(1+2), (12 + 2*5)/(1+2)) (x1, y1) = ((-3 + 6)/3, (2 + 10)/3) (x1, y1) = (3/3, 12/3) (x1, y1) = (1, 4)
Finding the Second Point
Now that we have found the coordinates of the first point, we can find the coordinates of the second point that divides the line segment in the ratio 2:1. Let the coordinates of this point be (x2, y2).
(x2, y2) = ((2*(-3) + 13)/(2+1), (22 + 1*5)/(2+1)) (x2, y2) = ((-6 + 3)/3, (4 + 5)/3) (x2, y2) = (-3/3, 9/3) (x2, y2) = (-1, 3)
Conclusion
In this problem, we found two points that divide the line segment joining A (3, 5) and B (-3, 2) into three equal parts. The coordinates of the first point are (1, 4) and the coordinates of the second point are (-1, 3).
Example
This problem can be used as an example to illustrate the concept of section formula. It can also be used to practice finding points that divide a line segment in a given ratio.
Key Concepts
- Section formula
- Points that divide a line segment in a given ratio
- Coordinates of points
Practice Problems
- Find two points that divide the line segment joining A (2, 3) and B (5, 7) into four equal parts.
- Find two points that divide the line segment joining A (-2, 4) and B (3, -1) into three equal parts.
Solutions
- The coordinates of the first point are (3, 5) and the coordinates of the second point are (4, 6).
- The coordinates of the first point are (-1, 3) and the coordinates of the second point are (1, 2).
Conclusion
In this article, we discussed how to find two points that divide the line segment joining A (3, 5) and B (-3, 2) into three equal parts. We used the concept of section formula to find the coordinates of the points. We also provided examples and practice problems to help readers understand the concept better.
Introduction
In our previous article, we discussed how to find two points that divide the line segment joining A (3, 5) and B (-3, 2) into three equal parts. We used the concept of section formula to find the coordinates of the points. In this article, we will provide a Q&A section to help readers understand the concept better.
Q&A
Q1: What is the section formula?
A1: The section formula is a formula used to find the coordinates of a point that divides a line segment in a given ratio. It is given by:
(x, y) = ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n))
Q2: How do I apply the section formula to find the coordinates of a point that divides a line segment in a given ratio?
A2: To apply the section formula, you need to know the coordinates of the two points that the line segment joins, and the ratio in which the point divides the line segment. You can then use the formula to find the coordinates of the point.
Q3: What is the difference between the section formula and the midpoint formula?
A3: The section formula is used to find the coordinates of a point that divides a line segment in a given ratio, while the midpoint formula is used to find the coordinates of the midpoint of a line segment. The midpoint formula is given by:
(x, y) = ((x1 + x2)/2, (y1 + y2)/2)
Q4: Can I use the section formula to find the coordinates of a point that divides a line segment in a ratio of 1:1?
A4: Yes, you can use the section formula to find the coordinates of a point that divides a line segment in a ratio of 1:1. In this case, the formula simplifies to:
(x, y) = ((x1 + x2)/2, (y1 + y2)/2)
Q5: How do I find the coordinates of the two points that divide a line segment in a given ratio?
A5: To find the coordinates of the two points that divide a line segment in a given ratio, you can use the section formula twice. First, you can find the coordinates of the point that divides the line segment in the given ratio, and then you can use the section formula again to find the coordinates of the other point.
Q6: Can I use the section formula to find the coordinates of a point that divides a line segment in a ratio of 2:1?
A6: Yes, you can use the section formula to find the coordinates of a point that divides a line segment in a ratio of 2:1. In this case, the formula is:
(x, y) = ((2x2 + 1x1)/(2+1), (2y2 + 1y1)/(2+1))
Q7: How do I know which point is the first point and which point is the second point?
A7: The first point is the point that divides the line segment in the given ratio, and the second point is the point that divides the line segment in the remaining ratio. For example, if the ratio is 1:2, the first point is the point that divides the line segment in the ratio 1:1, and the second point is the point that divides the line segment in the ratio 2:1.
Q8: Can I use the section formula to find the coordinates of a point that divides a line segment in a ratio of 3:2?
A8: Yes, you can use the section formula to find the coordinates of a point that divides a line segment in a ratio of 3:2. In this case, the formula is:
(x, y) = ((3x2 + 2x1)/(3+2), (3y2 + 2y1)/(3+2))
Q9: How do I know if the point I found using the section formula is the correct point?
A9: To know if the point you found using the section formula is the correct point, you can check if the point divides the line segment in the given ratio. You can do this by finding the coordinates of the two points that the line segment joins, and then using the section formula to find the coordinates of the point that divides the line segment in the given ratio.
Q10: Can I use the section formula to find the coordinates of a point that divides a line segment in a ratio of 4:3?
A10: Yes, you can use the section formula to find the coordinates of a point that divides a line segment in a ratio of 4:3. In this case, the formula is:
(x, y) = ((4x2 + 3x1)/(4+3), (4y2 + 3y1)/(4+3))
Conclusion
In this article, we provided a Q&A section to help readers understand the concept of the section formula and how to use it to find the coordinates of a point that divides a line segment in a given ratio. We also provided examples and practice problems to help readers understand the concept better.